Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)_n$ is the nerve of the category of functors $[n]\to C$ and natural weak equivalences between them. If we fibrantly replace $N(C,W)$ in the complete Segal space model structure, we obtain an $(\infty,1)$-category. Is this $(\infty,1)$-category equivalent to the $(\infty,1)$-categorical localization of $C$ at $W$ (e.g. the simplicial hammock localization, or a fibrant replacement of its "marked simplicial nerve")?

I believe that when $C$ is a model category, this follows from results in the various papers of Dwyer and Kan on simplicial localization. It could be that the same is true for the general case, but I haven't yet been able to see how.

I have been asking me this as well a while ago. I think for it is important to figure out to which extend for the given pair (C,W) and a small category I the functor category Hom(I,C) with the induced notion of weak equivalences models all maps of infinity categories $NI \to L(C,W)$. This is true for a properly behaved model category, but in general I have no idea.
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Thomas NikolausApr 2 '12 at 20:56

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It seems unlikely to me that Hom(I,C) could model all $(\infty,1)$-functors $I\to L(C,W)$, since you haven't applied any fibrant replacement to $(C,W)$ to make the morphisms in $W$ into equivalences.
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Mike ShulmanApr 2 '12 at 21:42

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I think the result you need is in a paper of Barwick and Kan. They show that the functor N(C,W) is half of a Quillen equivalence.
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Jeff SmithApr 4 '12 at 6:41

@Jeff, that paper was also suggested in another answer that's since been deleted. Can you explain why knowing that N(C,W) is half of a Quillen equivalence also tells you that it is equivalent to the localization of C at W, without some sort of additional argument like the one Denis-Charles gave?
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Mike ShulmanApr 4 '12 at 7:12

2 Answers
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Let us fix some notations. I will write $i(X)$ for the maximal Kan subcomplex of a quasi-category $X$ and $Hom(A,B)$ for the internal Hom of simplicial sets $A$ and $B$. If $B$ is a quasi-category, then so is $Hom(A,B)$ for any simplicial set $A$, and I will write
$$i(A,B)=i(Hom(A,B))$$
It is a fact that (at least up to simplicial homotopy) any complete Segal space is of the form $i(\Delta^\ast,X)$ for a quasi-category $X$, where $\Delta^\ast$ is the standard cosimplicial object of simplicial sets (given by the Yoneda embedding); furthermore, the canonical inclusion $X\to i(\Delta^\ast,X)$ is a weal equivalence (where $X$ is considered as a (constant) simplicial quasi-category). Therefore, it is sufficient to understand morphisms of bisimplicial sets $N(C,W)\to i(\Delta^\ast,X)$ in the homotopy category of the Rezk model structure. The point is that these are homotopy classes of maps of bisimplicial sets (being maps from a cofibrant object to a fibrant one), and that those morphisms of bisimplicial sets can be understood rather explicitely in terms of morphisms of marked simplicial sets.

If $u:N(C)\to i(\Delta^\ast,X)$ is a morphism of bisimplicial sets, it is completely
determined by the morphism of simplicial sets $u_0:N(C)\to X=i(\Delta^\ast,X)_0$. On the other hand, the map $u$ sends any arrow of $W$ to an invertible $1$-simplex of $X$ if and only if the maps
$$N(Hom([n],C))=Hom(\Delta^n,N(C))\to Hom(\Delta^n,X)$$
induced by $u_0$ define maps
$$N(C,W)_n\to i(\Delta^n,X)$$
which in turn define a morphism of bisimplicial sets
$$N(C,W)\to i(\Delta^\ast,X)$$
Moreover, any map $N(C,W)\to i(\Delta^\ast,X)$ is obtained in this way (I guess that interpreting all this in terms of marked simplicial sets would help to write down all this in a cleaner way). It is then easy to deduce from there that $N(C,W)$ has the universal property of the localization of $N(C)$ by $N(W)$ in the homotopy category of the Rezk model structure: to deal with homotopies, you just have to replace $C$ by $C\times I$, where $I$ denotes the contractible groupoid with two objects (with an adequate subcategory of weak equivalences).

This looks like a promising approach, but I don't follow it yet. When you write $N(C)$, you mean to regard it as a bisimplicial set discrete in which direction? And it seems to me that $i(\Delta^0,X) = i(X)$, not $X$.
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Mike ShulmanApr 3 '12 at 0:03

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I see quasi-categories as constant simplicial quasi-categories. Also, for a quasi-category $X$, I see $i(\Delta^\ast,X)$ as a simplicial object in the category of quasi-categories: for any fixed integer $q$, $i(\Delta^\ast,X)_q$ is a quasi-category (and for $q=0$, this is precisely $X$).
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Denis-Charles CisinskiApr 3 '12 at 11:48

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Okay, I see! How about the following for a reformulation in terms of marked simplicial sets? Your functor $i(\Delta^*,-)$ from quasicategories to complete Segal spaces induces an equivalence of homotopy categories (it is homotopy equivalent to the functor $k^!$ from arxiv.org/abs/math.AT/0607820). But $i(\Delta^*,-)$ also factors through the functor $X\mapsto (X,i(X)_1)$ from quasicategories to marked simplicial sets, which also induces an equivalence of homotopy categories. ...
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Mike ShulmanApr 3 '12 at 19:39

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... The factorization is via the functor $G$ from marked simplicial sets to bisimplicial sets defined by $G(X)_{n,m} = MSSet((\Delta^n)^\flat \times (\Delta^m)^\sharp, X)$; thus $G$ also induces an equivalence of homotopy categories. (In fact, it appears that $G$ is even part of a Quillen equivalence.) But $G$ takes the "marked simplicial nerve" of $(C,W)$ to the bisimplicial set $N(C,W)$, and the marked simplicial nerve clearly represents the localization of $C$ at $W$.
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Mike ShulmanApr 3 '12 at 19:39

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Your reformulation is quite nice. Moreover, all this shows that you can replace $N(C)$ by any quasi-category.
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Denis-Charles CisinskiApr 3 '12 at 21:23

Yes, this follows easily by combining the results of Barwick-Kan and Toen. One way to rephrase your question is the following:

Given a relative category $(C,W)$ (i.e. just a category with a subcategory weak equivalences containing the identities) is the classification diagram $N(C,W)$ weakly equivalent to $N^{CSS}(L^H(C,W))$ in Rezk's model category of complete Segal spaces?

Here $N^{CSS}$ denotes any homotopical functor (i.e. preserving weak equivalences) from simplicial categories to complete Segal spaces implementing the equivalence of homotopy theories. For example (and for definiteness) we can take $N^{CSS}$ to be the composite of the (derived) homotopy coherent nerve of Cordier (from simplicial categories to quasicategories) and the right Quillen equivalence from quasicategories to complete Segal spaces (denoted $t^!$ in Joyal-Teirney's "Quasi-categories vs. Segal Spaces"). You will see that it doesn't really matter much what precisely we use here, just that it induces a reasonable equivalence of homotopy theories.

In otherwords we are asking if the classification diagram is equivalent to what you get by transforming the Hammock localization into a complete Segal space.

Now in Barwick and Kan's paper "relative categories: another model for the homotopy theory of homotopy theories" they show among other things:

There is a model category structure on the category RelCat of relative categories;

There is a Quillen equivalence of this model structure with the model structure of complete Segal spaces;
$$ K_\xi: CSS \leftrightarrows RelCat: N_\xi$$

There is natural map $N(C,W) \to N_\xi(C,W)$ which is a weak equivalence in the CSS model structure.

Thus to answer your question it is enough to show that $N_\xi(C,W)$ is equivalent to $N^{CSS}(L^H(C,W))$ in the homotopy category of complete Segal spaces.

Next, in Barwick and Kan's paper "In the category of relative categories the equivalences are exactly the DK-equivalences" they show that

The hammock localization $L^H$ is an equivalence from the relative category of relative categories to the relative category of simplicial categories (each using the weak equivalences from their respective model structures).

So now we are in the situation where we know that the functors $N^{CSS} \circ L^H$ and $N_\xi$ each induce equivalences from the homotopy category of Relative categories to the homotopy category of complete Segal spaces. (In fact they lift to equivalences in the model category of (say) relative categories).

We want to know if they induce the same equivalence of homotopy categories. But this is answered precisely by Toen's theorem (in "Vers une axiomatisation de la theorie des categories superieures"), which says that up to equivalence there are precisely two such equivalences. They are distinguished by what happens to the subcategory consisting of the terminal category and the free walking arrow. This subcategory is clearly preserved by both $N^{CSS} L^H$ and $N_\xi \simeq N$. Hence we conclude they give rise the the same equivalences of homotopy categories, and hence

In short, Barwick and Kan show that $N^{CSS} L^H$ and $N$ induce equivalences of homotopy theories, Toen shows that up to equivalence there are only two possibilities for such equivalences, and a simple calculation shows that in fact they agree.

Ah, excellent! I had thought of using Toen's result, but I didn't realize that Barwick-Kan had also proved that hammock localization was an equivalence of homotopy theories. Too bad I can only accept one answer.
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Mike ShulmanApr 4 '12 at 17:33

Does this approach extend to "relative quasicategories" as well?
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Mike ShulmanApr 4 '12 at 17:33

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Hmm, and in fact the statement I wanted is more or less explicit in a 2009 draft of Barwick-Kan that I had sitting around, but which doesn't seem to be available any more, called "Relative categories; another model for the homotopy theory of homotopy theories part II: the weak equivalences".
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Mike ShulmanApr 4 '12 at 17:50

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@Mike: "Does this approach extend to "relative quasicategories" as well?" I imagine it will, but it won't be out-of-the-box like your original question. I know that in one of their papers Barwick and Kan consider an analog of some of this structure for relative simplicial categories.
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Chris Schommer-PriesApr 4 '12 at 19:10